TY - JOUR

T1 - Colorful Strips

AU - Aloupis, Greg

AU - Cardinal, Jean

AU - Collette, Sébastien

AU - Imahori, Shinji

AU - Korman, Matias

AU - Langerman, Stefan

AU - Schwartz, Oded

AU - Smorodinsky, Shakhar

AU - Taslakian, Perouz

PY - 2011/5/1

Y1 - 2011/5/1

N2 - We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k-1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k-1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

AB - We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k-1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k-1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

KW - Computational geometry

KW - Covering decomposition

KW - Hypergraph coloring

KW - Lovász local lemma

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U2 - 10.1007/s00373-011-1014-5

DO - 10.1007/s00373-011-1014-5

M3 - Article

AN - SCOPUS:79954633707

SN - 0911-0119

VL - 27

SP - 327

EP - 339

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 3

ER -